सिद्ध कीजिए कि
` cos[tan^(-1){sin(cot^(-1)x)}]=sqrt((x^(2)+1)/(x^(2)+2))`

Explanation

सर्वप्रथम माना `cot^(-1)x=theta`
`implies x= cot theta `
`:. " cosec" theta=sqrt(1+cot^(2)theta)=sqrt(1+x^(2))`
` implies sin theta = (1)/(sqrt(1+x^(2)))`
`implies sin (cot^(-1)x)=(1)/(sqrt(1+x^(2)))`
`= tan^(-1)[sin(cot^(-1)x)]=tan^(-1)""(1)/(sqrt(1+x^(2)))= phi` (say)
तब `, cos[tan^(-1){sin(cot^(-1)x)}]=cos phi" "`......(1)
अब , चूँकि `tan^(-1)""(1)/(sqrt(1+x^(2)))=phi` ,
तब ` tan phi=(1)/(sqrt(1+x^(2)))`
`:. sec phi=sqrt(1+tan^(2)phi)=sqrt(1+(1)/((1+x^(2))))=sqrt((x^(2)+1)/(x^(2)+2))`
`:. cos phi= sqrt((1+x^(2))/(2+x^(2)))" " `.....(2)
अतः समीकरण ( 1 ) व ( 2 ) से ,
` cos[tan^(-1){sin(cot^(-1)x)}]=sqrt(x^(2)+1)/(x^(2)+2)`